Spaces of AANRs

Abstract

For a metric space $X$, let ${\text {AAN}}{{\text {R}}_{\text {C}}}(X)$ denote the hyperspace of all nonempty approximative absolute neighborhood retracts in the sense of Clapp in $X$ topologized with the metric of continuity. We show that ${\text {AAN}}{{\text {R}}_{\text {C}}}(X)$ is topologically complete iff $X$ is topologically complete. Some subsets of the first Baire category in ${\text {AAN}}{{\text {R}}_{\text {C}}}(X)$ for a $Q$-manifold $X$ are identified. For example, the collection ${\text {AAN}}{{\text {R}}_{\text {N}}}(X)$ of all nonempty approximative absolute neighborhood retracts in the sense of Noguchi in $X$ is such a subset.